Dee Capacitance & RF Matching Calculator

The math

Capacitance estimate

A dee of radius r presents a half-disc face of area A = πr²/2 toward the liner above it and again below it — two parallel-plate capacitors in parallel:

The fringe factor (default 20%) crudely accounts for edge fields, and the stem allowance stands in for the dee stem and feed structure — both defaults are placeholders to adjust, not measured constants. This is deliberately a crude model: it omits the accelerating-gap, support, and liner contributions, which for some geometries exceed the terms it keeps, so the real value can differ by tens of percent or more. Measure it (grid-dip a known inductor against the dee, or use a VNA) before winding a final resonator coil.

Resonance

At resonance the dee tank looks, from its feed point, like a pure resistance R_p — the parallel loss resistance set by conductor losses (R_p = Q_tank · X_C, where Q_tank is the unloaded tank Q: measure it with the drive and matching network disconnected or de-embedded, because a Q read through the coupling is the loaded value and gives the wrong R_p and power estimate). The scale is then plain arithmetic: at this page's worked values X_C ≈ 650 Ω, so tank Qs from a few tens (mechanical joints, thin plating) to a few hundred (clean brazed copper) put R_p anywhere from several kΩ past 100 kΩ. Measure yours.

Matching 50 Ω to R_p

With R_p > 50 Ω, the canonical L-network places the series element on the low-resistance (source) side and the shunt element across the high-resistance (dee) side. This calculator uses the high-pass form — series capacitor, shunt inductor — which also blocks DC and, where the coil arrangement really does give the dee a continuous DC path to ground, drains static charge off it. Treat that as a circuit convenience: verify the path exists in your build, and it is never a substitute for the discharge and grounding practices on the safety page:

In practice L_p and the resonating inductance L are often built as one tapped coil, but the two displayed values do not carry over directly: a tapped resonator is an autotransformer with its own equivalent-circuit synthesis, tuned on the bench — and naively paralleling the two displayed inductors gives L∥L_p, which resonates at the wrong frequency. The two-element form is kept because its algebra is transparent. Component values are exact only at f, and the 1/Q fractional bandwidth below is the matching network's: a rule of thumb for how sharply the match degrades off-frequency, with Q the transformation Q above. The measured −3 dB bandwidth of dee voltage is a different quantity, set by the loaded tank Q, the coupling, component losses, and any beam loading.

Assumptions and limits

• Parallel-plate capacitance with a flat fringe factor — no account of dee thickness, aperture, pillars, or the accelerating-gap capacitance to the dummy dee.
• R_p is an input, not a prediction; it varies with construction quality, plating, and joint resistance, and it sets the drive power for a given dee voltage: P = V_dee²/2R_p, with V_dee the peak RF amplitude of the dee relative to ground (the convention this site's energy-gain figures use). An RMS, peak-to-peak, or dee-to-dee number changes the result by a factor of 2–4, so fix the convention before comparing.
• Ideal lossless matching components; at tens of kΩ transformation ratios, real component Q and stray reactance matter — and so do ratings: the series capacitor and shunt coil carry RF voltages and circulating currents far above the 50 Ω drive level, so component voltage and current stress, heating, and derating are a separate, safety-relevant design step this page does not compute.
• Beam loading is ignored. Its scale is beam current times final energy in volts: a 100 nA beam at 150 keV is 15 mW and genuinely negligible, while tens of microamps at hundreds of keV are watts to tens of watts, comparable to tank loss in a clean small system — include it once currents grow.

Worked check

Dee radius 13 cm, 25 mm gaps top and bottom, 20% fringe, 5 pF stem: face area 265 cm², C_dee ≈ 27.6 pF. At 8.87 MHz (protons at 0.582 T) the resonating inductance is 11.7 µH. Matching 50 Ω to R_p = 20 kΩ: Q = 20.0, C_s = 18.0 pF, L_p = 18.0 µH, match bandwidth ≈ 440 kHz.

Sources

• Parallel-plate capacitance: any electromagnetics text, e.g. D. J. Griffiths, Introduction to Electrodynamics, §2.5.
• L-network design equations: C. Bowick, RF Circuit Design, 2nd ed., Newnes, 2008, ch. 4; ARRL Handbook, impedance matching chapter.
• Cyclotron RF systems: J. J. Livingood, Principles of Cyclic Particle Accelerators, Van Nostrand, 1961 — ch. 12.

Related sourced design rules: the design guide’s RF + dee rules. For the architecture and coupling decisions behind these numbers — self-excited vs driven, loop vs capacitive tap, multipactor, dee-voltage measurement — see Driving the Dee: RF Coupling.